IEEE Trans Neural Netw Learn Syst. 2017 Dec;28(12):3032-3044. doi: 10.1109/TNNLS.2016.2614130. Epub 2016 Oct 10.
Generalized eigendecomposition problem has been widely employed in many signal processing applications. In this paper, we propose a unified and self-stabilizing algorithm, which is able to extract the first principal and minor generalized eigenvectors of a matrix pencil of two vector sequences adaptively. Furthermore, we extend the proposed algorithm to extract multiple generalized eigenvectors. The performance analysis shows that only the desired equilibrium point of the proposed algorithm is stable and all others are (unstable) repellers or saddle points. Convergence analysis based on the deterministic discrete-time approach shows that, for a step size within a certain range, the norm of the principal/minor state vector converges to a fixed value that relates to the corresponding principal/minor generalized eigenvalue. Thus, the proposed algorithm is a generalized eigenpairs (eigenvectors and eigenvalues) extraction algorithm. Finally, the simulation experiments are carried to further demonstrate the efficiency of the proposed algorithm.
广义特征分解问题在许多信号处理应用中得到了广泛应用。在本文中，我们提出了一种统一的自稳定算法，能够自适应地提取两个向量序列的矩阵束的第一主广义特征向量和次广义特征向量。此外，我们将所提出的算法扩展到提取多个广义特征向量。性能分析表明，只有所提出算法的期望平衡点是稳定的，而其他所有平衡点都是（不稳定的）排斥点或鞍点。基于确定性离散时间方法的收敛性分析表明，对于一定范围内的步长，主/次状态向量的范数收敛到一个与相应的主/次广义特征值相关的固定值。因此，所提出的算法是一种广义特征对（特征向量和特征值）提取算法。最后，进行了仿真实验以进一步验证所提出算法的有效性。
